Área e volume da esfera

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ❚❡rr❛

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚

➪r❡❛ ❡ ❱♦❧✉♠❡ ❞❛ ❊s❢❡r❛

♣♦r

▲❡♦♥❛r❞♦ ❆♥❞r❛❞❡ ▼❡❞❡✐r♦s

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❋❛❣♥❡r ▲❡♠♦s ❞❡ ❙❛♥t❛♥❛

❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊❚✲❯❋❘◆✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜✲ t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

▼❛rç♦✴✷✵✶✹ ◆❛t❛❧ ✲ ❘◆

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s✱ ♣♦r ♠❡ ❝♦♥❢♦rt❛r ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❛ ❝❛♠✐♥❤❛❞❛ ❡ ♠❡ ❞❛r ❢♦rç❛ ✐♥t❡r✐♦r ♣❛r❛ s✉♣❡r❛r ♦s ♦❜stá❝✉❧♦s✱ ♠♦str❛r ♦s ❝❛♠✐♥❤♦s ♥❛s ❤♦r❛s ✐♥❝❡rt❛s ❡ s✉♣r✐r t♦❞❛s ❛s ♠✐♥❤❛s ♥❡❝❡ss✐❞❛❞❡s✳

❆♦ ♠❡✉ ♣❛✐✱ ✐r♠ã♦s✱ ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s✱ ♣❡❧♦ ❡stí♠✉❧♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳

❆♦ Pr♦❢❡ss♦r ❋❛❣♥❡r ▲❡♠♦s ❞❡ ❙❛♥t❛♥❛✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ❝♦❧❛❜♦r❛çã♦✱ ❝♦✲ ♥❤❡❝✐♠❡♥t♦s tr❛♥s♠✐t✐❞♦s ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡stí♠✉❧♦ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✳ ❖❜r✐❣❛❞♦ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡✳

➚ ❈♦♦r❞❡♥❛❞♦r❛ ❞♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✲ P❘❖❋▼❆❚✴❈❈❊❚✲❯❋❘◆✱ Pr♦❢❡ss♦r❛ ❱✐✈✐❛♥❡ ❙✐♠✐♦❧✐ ▼❡❞❡✐r♦s ❈❛♠♣♦s ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ♥✉♠ s♦♥❤♦ q✉❡ ❛❣♦r❛ é ❞❡ t♦❞♦s✳ ❈♦♠ ❞❡❞✐❝❛çã♦ ❡ ❝♦♠♣❡tê♥❝✐❛ ❡❧❛ ❝♦♥❞✉③✐✉ ❡ss❡ ♠❡str❛❞♦ ❞✉r❛♥t❡ ♦s ú❧t✐♠♦s três ❛♥♦s✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✴❈❈❊❚✲❯❋❘◆ q✉❡ ❞❡s❡♠♣❡♥❤❛r❛♠ ❝♦♠ ❞❡❞✐❝❛✲ çã♦ ❛s ❛✉❧❛s ♠✐♥✐str❛❞❛s✳

➚ ♣r♦❢❡ss♦r❛ ◆❛t❤❛❧✐❡ ❇r❛❣❛✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ q✉❡ s❡♠♣r❡ ❞❡✲ ♠♦♥str♦✉✱ ♣❡❧❛s ❝♦rr❡çõ❡s ♦rt♦❣rá✜❝❛s✱ ♣❡❧❛s ❛❥✉❞❛s ♥❛ ❧❡✐t✉r❛ ❞♦ tr❛❜❛❧❤♦✱ ♣❡❧❛s ❜♦❛s s✉❣❡stõ❡s ❡ ♣❡❧♦ r✐❣♦r ❝✐❡♥tí✜❝♦ q✉❡ ❧❤❡ é ❝❛r❛❝t❡ríst✐❝♦ ❡ q✉❡ ❡♠ ♠✉✐t♦ ✈❡✐♦ ❡♥r✐q✉❡❝❡r ♦ tr❛❜❛❧❤♦✳

❆♦ ♠❡✉ ❣r❛♥❞❡ ❛♠✐❣♦ ❊♠❛♥✉❡❧ ●♦♠❡s ♣❡❧❛ ❝♦❧❛❜♦r❛çã♦ ♥❛ ♣❡sq✉✐s❛ ❞❡ss❡ tr❛❜❛✲ ❧❤♦ ❡ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠ ❡ ♠❡ ✐♥❝❡♥t✐✈❛r ♣♦✐s s❛❜✐❛ q✉❡ ❡✉ ✐r✐❛ ❝♦♥s❡❣✉✐r ❛❧❝❛♥ç❛r ❡ss❡ ♦❜❥❡t✐✈♦✳

❆♦ s❡♠♣r❡ ♣r❡st❛t✐✈♦ ❡ ❛♠✐❣♦ ❋r❛♥❝✐s❝♦ ❆❧❞r✐♥ ❘✉✜♥♦ ❆r♠str♦♥❣✱ ♣❡❧❛ ❛❥✉❞❛ ♥♦ ▲❆❚❊❳ ❡ ♥❛ ❡❧❛❜♦r❛çã♦ ❞❛ ❛♣r❡s❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❖❜r✐❣❛❞♦ ♣♦r s❡♠♣r❡ t❡r ♠❡ ❛t❡♥❞✐❞♦ q✉❛♥❞♦ ❡✉ ♣r❡❝✐s❡✐✳

❆♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ■❋❘◆ ❡ P❘❖❋▼❆❚✴❯❋❘◆✲❯❋P❇✲❯❋❊❘❙❆✱ ❙❛♥❞r♦ ●♦✲ ❞❡✐r♦✱ ❚❤✐❛❣♦ ❱❛❧❡♥t✐♠✱ ❋❡❧❧✐♣❡ ❆rr❛✐s✱ ▲✉❝✐❛♥♦ ◆ó❜r❡❣❛✱ ◆✐❧s♦♥ ❍❡r♠í♥✐♦✱ ●✐❧❜❡rt♦

❋❡r♥❛♥❞❡s✱ ❏♦r❣❡ P♦♥t❡s ❡ ❆♥tô♥✐♦ ❘♦❜❡rt♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❛♠✐③❛❞❡ ❡ ❡st✉✲ ❞♦s ❞✉r❛♥t❡ ♦s ú❧t✐♠♦s ❛♥♦s✳

❆♦ Pr♦❢❡ss♦r ▼❛①②♠♠❡ ▼❡❧♦✱ ♣❡❧❛ ❝♦rr❡çã♦ ❞♦ ❆❇❙❚❘❆❈❚ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦s ♠❡✉s ❛❧✉♥♦s ❞❛ ▲✐❝❡♥❝✐❛t✉r❛ ❡♠ ▼❛t❡♠át✐❝❛ ❞♦ ■❋❘◆✴◆❛t❛❧✲❈❡♥tr❛❧ ♣❡❧♦ ❛♣♦✐♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ♦ ❡♥s✐♥♦ ❞❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ❝á❧✲ ❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❡ ❞❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳ ❙♦❜r❡ ❡st❡ t❡♠❛✱ ❢♦✐ ❢❡✐t❛ ✉♠❛ ♣❡sq✉✐s❛ ❞❡ ❝❛rát❡r q✉❛♥t✐✲q✉❛❧✐t❛t✐✈♦ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ✐❞❡♥t✐✜❝❛r ❞❡ q✉❡ ❢♦r♠❛ ❡st❡s t❡♠❛s sã♦ ❛❜♦r❞❛❞♦s✳ P❛r❛ ✐ss♦✱ ✜③❡♠♦s ✶✹ ♣❡r❣✉♥t❛s ❛ ✸✵ ♣r♦❢❡ss♦r❡s ❞❡ ◆❛t❛❧ ❡ ♦s r❡s✉❧t❛❞♦s ❞❡st❛ ♣❡sq✉✐s❛ sã♦ ❛♣r❡s❡♥t❛❞♦s ❡ ❞✐s❝✉t✐❞♦s✳ ❉❡♣♦✐s ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ❢♦r♠❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❞❡❞✉③✐r ❛s ❢ór♠✉❧❛s ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❡ ❞❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✳ ➪r❡❛s✳ ❱♦❧✉♠❡s✳ ❊s❢❡r❛✳ ❊♥s✐♥♦ ▼é❞✐♦✳

❆❜str❛❝t

■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥✱ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ♦♥ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ✈♦❧✉♠❡ ♦❢ t❤❡ s♣❤❡r❡ ❛♥❞ t❤❡ ❛r❡❛ ♦❢ s♣❤❡r✐❝❛❧ s✉r❢❛❝❡✳ ❖♥ t❤✐s t♦♣✐❝✱ ❛ q✉❛❧✐✲q✉❛♥t✐t❛t✐✈❡ ✇❛s t❛❦❡♥ s✉r✈❡② ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ ✐❞❡♥t✐❢②✐♥❣ ❤♦✇ t❤❡s❡ t♦♣✐❝s ❛r❡ ❛❞❞r❡ss❡❞✳ ❋♦r t❤✐s✱ ✇❡ ♠❛❞❡ ✶✹ q✉❡st✐♦♥s t♦ ✸✵ t❡❛❝❤❡rs ♦❢ ◆❛t❛❧ ❛♥❞ t❤❡ r❡s✉❧ts ♦❢ t❤✐s s✉r✈❡② ❛r❡ ♣r❡s❡♥t❡❞ ❛♥❞ ❞✐s❝✉ss❡❞✳ ❆❢t❡r t❤❛t✱ ✇❡ ♣r❡s❡♥t ❛❧t❡r♥❛t✐✈❡ ✇❛②s t♦ ❞❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ ✈♦❧✉♠❡ ♦❢ ❛ s♣❤❡r❡ ❛♥❞ t❤❡ ❛r❡ ♦❢ ❛ s♣❤❡r✐❝❛❧ s✉r❢❛❝❡✳

❑❡②✇♦r❞s✿ ❙♣❛t✐❛❧ ●❡♦♠❡tr②✳ ❙♣❤❡r❡ ❆r❡❛✳ ❱♦❧✉♠❡✳ ❍✐❣❤ ❙❝❤♦♦❧✳

❙✉♠ár✐♦

❉❡❞✐❝❛t➹3

ria ✈

✶ ■♥tr♦❞✉çã♦ ✶

✷ P❡sq✉✐s❛✿ ❈♦♠♦ ♦s ♣r♦❢❡ss♦r❡s ❛❜♦r❞❛♠ ♠❡❧❤♦r ♦ ✈♦❧✉♠❡ ❡ ❛ ár❡❛

❞❛ ❡s❢❡r❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛❄ ✸

✷✳✶ ❉❛❞♦s ❞❛ P❡sq✉✐s❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✶✳✶ ❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✸ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❡ ❈♦♥❝❡✐t♦s ●❡♦♠étr✐❝♦s ✶✷ ✸✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✷ ❆ ◆♦çã♦ ■♥t✉✐t✐✈❛ ❞❡ ❱♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✸ ❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✸✳✹ ❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹ ➪r❡❛s ❡ ❱♦❧✉♠❡s ❯s❛♥❞♦ ❆♣r♦①✐♠❛çõ❡s ✷✹ ✹✳✶ ❱♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r ❝✐❧✐♥❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✹✳✷ ➪r❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r r❡tâ♥❣✉❧♦s ✳ ✳ ✸✼

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✸

❈❛♣ít✉❧♦ ✶

■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛♠♦s ♦ ❡♥s✐♥♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❛ ❡s❢❡r❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❖ ♣r✐♥❝✐♣❛❧ ♣♦♥t♦ é ❛♣r❡s❡♥t❛r ♠❛♥❡✐r❛s ❞❡ ❞❡❞✉③✐r ❛s r❡s♣❡❝t✐✈❛s ❢ór♠✉❧❛s q✉❡ ♣♦ss❛♠ s❡r tr❛❜❛❧❤❛❞❛s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ s❡rã♦ ♠♦str❛❞♦s ❡ ❞✐s❝✉t✐❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ✉♠❛ ♣❡sq✉✐s❛ ❛♣❧✐❝❛❞❛ ❝♦♠ ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡ ◆❛t❛❧ ❛❝❡r❝❛ ❞❡ ❝♦♠♦ é ❢❡✐t❛ ❛ ❛❜♦r❞❛❣❡♠ ❞♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❉❡st❛❝❛♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ♥♦s ❢❛♠✐❧✐❛r✐③❛r♠♦s ❝♦♠ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s✱ ♣♦✐s ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞❡ss❛s ♣❡r❝❡♣çõ❡s ❡ ❤❛❜✐❧✐❞❛❞❡s ❞❡ ✐❞❡♥t✐✜❝❛r♠♦s ❛s ✜❣✉r❛s sã♦ ❡ss❡♥❝✐❛✐s ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s ❡s❝♦❧❛r❡s ❡ ❞♦ ❝♦t✐❞✐❛♥♦✱ ♥♦s ❛✉①✐❧✐❛♥❞♦ ❛ss✐♠ ❛ ❛♣r❡♥❞❡r ♦ s✐❣♥✐✜❝❛❞♦ ❡ ❛s ❢✉♥çõ❡s ❞♦ ❡♥s✐♥♦ ❞❛ ❣❡♦♠❡tr✐❛✳ ❖ ✉s♦ ❞❡ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦ t❛♠❜é♠ é ❞❡ s✉♠❛ r❡❧❡✈â♥❝✐❛✱ ♣♦✐s ♣♦ss✐❜✐❧✐t❛ ❛♦s ❛❧✉♥♦s r❡❧❛❝✐♦♥❛r ❝♦♥t❡ú❞♦s tr❛❜❛❧❤❛❞♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ❛ s✉❛ ✈✐✈ê♥❝✐❛ s♦❝✐❛❧✳ P♦r ✜♠✱ ♠♦str❛r❡♠♦s q✉❡ ❛t✉❛❧♠❡♥t❡ ❛s ❞❡♠♦♥str❛çõ❡s ♠❛t❡♠át✐❝❛s ♠✉✐t❛s ✈❡③❡s ❡stã♦ s❡♥❞♦ ❞❡✐①❛❞❛s ❞❡ ❧❛❞♦✱ ♦ q✉❡ ♥♦s ❢❛③ ♣❡♥s❛r q✉❡ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ♥ã♦ ❜✉s❝❛♠ ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❛❧✉♥♦s ❡ s✐♠♣❧❡s♠❡♥t❡ ✏❥♦❣❛♠✑ ❛s ❢ór♠✉❧❛s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ r❡❧❛t❛♠♦s ❛❧❣✉♥s ❛s♣❡❝t♦s ❤✐stór✐❝♦s ❞❛ ❣❡♦✲ ♠❡tr✐❛✱ ❝♦♠ ❞❡st❛q✉❡ ♣❛r❛ ❛ ❣❡♦♠❡tr✐❛ ❞♦s ❡❣í♣❝✐♦s ❡ ❞♦s ❣r❡❣♦s✳ ❇✉s❝❛♠♦s ❞❡✜♥✐r ✉♠ ❝♦♥❝❡✐t♦ ♠❛✐s ❝♦♥❝r❡t♦ ♣❛r❛ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ q✉❛❧q✉❡r s❡♥❞♦ ❡❧❡ r❡❣✉❧❛r ♦✉ ♥ã♦✱ ❛ ♣❛rt✐r ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛ ❞❡ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ♦✉ ♣♦r ❝✐❧✐♥❞r♦s✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛♠♦s q✉❡ ♣❛r❛ ♠❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛ q✉❛❧q✉❡r s❡♠♣r❡ ❞❡✈❡♠♦s ❝♦♠♣❛r❛r ❝♦♠ ♦✉tr❛ ❝♦♥❤❡❝✐❞❛ ❡ ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♥✐❞❛❞❡ q✉❡ s❡r✈❡ ❞❡ r❡❢❡rê♥❝✐❛ ♣❛r❛ t❛❧ ♠❡❞✐çã♦✳ ❈♦♠ ✐ss♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ✈♦❧✉♠❡✳ P♦r ✜♠✱ ✉s❛♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ❞❡❞✉③✐r ❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳

❊st❛ ❞❡❞✉çã♦ ✉s❛♥❞♦ ❈❛✈❛❧✐❡r✐ é ❛ ♠❛✐s ✉s❛❞❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳

◆♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❛ ✜♠ ❞❡ q✉❡ ♣♦ss❛♠♦s ❛♣❧✐❝❛r ❛s ❞❡♠♦♥str❛çõ❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❞❡s❝r❡✈❡r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❡♥t❡♥❞✐✲ ♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❞❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳ ❊♠ s❡❣✉✐❞❛✱ ♠♦str❛r❡♠♦s ✉♠❛ ♠❛♥❡✐r❛ ✐♥t❡r❡ss❛♥t❡ ✭❡ ❞✐❢❡r❡♥t❡✮ ❞❡ ♦❜t❡r ❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳ ❊st❛ ❢♦r♠❛ ❞❡ ♦❜t❡r ❛ ❢ór♠✉❧❛ ✉s❛ ❝♦rt❡s tr❛♥s✈❡rs❛✐s ❡♠ ✉♠❛ s❡♠✐✲❡s❢❡r❛✱ ♦s q✉❛✐s ❞❡✲ t❡r♠✐♥❛♠ ❝✐❧✐♥❞r♦s ✐♥s❝r✐t♦s ♥❛ ❡s❢❡r❛✳ ❉❡♣♦✐s✱ ✉s❛♠♦s ♦s ✈♦❧✉♠❡s ❞❡st❡s ❝✐❧✐♥❞r♦s ✐♥s❝r✐t♦s ❝♦♠♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳ ❊st❛ ✐❞é✐❛ ❞❡ ❛♣r♦①✐✲ ♠❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ❝♦♠♦ ✉♠ ♣r✐♠❡✐r♦ ❝♦♥t❛t♦ ❞♦ ❛❧✉♥♦ ❝♦♠ ❛ ✐❞é✐❛ ✐♥t✉✐t✐✈❛ ❞❡ ❧✐♠✐t❡✱ ❝♦♥❝❡✐t♦ s♦♠❡♥t❡ ✈✐st♦ ♥♦s ❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✳ ❉❡♣♦✐s✱ ✉s❛♥❞♦ t❛♠❜é♠ ❛ ✐❞é✐❛ ❞❡ ❛♣r♦①✐♠❛çõ❡s✱ ❢❛③❡♠♦s ❛ ❞❡❞✉çã♦ ❞❛ ❢ór♠✉❧❛ ♣❛r❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳

❈❛♣ít✉❧♦ ✷

P❡sq✉✐s❛✿ ❈♦♠♦ ♦s ♣r♦❢❡ss♦r❡s

❛❜♦r❞❛♠ ♠❡❧❤♦r ♦ ✈♦❧✉♠❡ ❡ ❛ ár❡❛

❞❛ ❡s❢❡r❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛❄

✷✳✶ ❉❛❞♦s ❞❛ P❡sq✉✐s❛

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛ ♣❡sq✉✐s❛ q✉❛♥t✐✲q✉❛❧✐t❛t✐✈❛✱ ❢✉♥❞❛♠❡♥t❛❞❛ ❡♠ ✉♠ q✉❡st✐♦♥ár✐♦ ❝♦♠♣♦st♦ ❞❡ q✉❡stõ❡s ♦❜❥❡t✐✈❛s ❡ s✉❜❥❡t✐✈❛s✱ tr❛ç♦✉ ♦ ♣❡r✜❧ ❞❡ ✸✵ ❡♥tr❡✈✐st❛❞♦s ❡ ✐❞❡♥t✐✜❝♦✉ ❝♦♠♦ sã♦ ❛❜♦r❞❛❞♦s ♦ ✈♦❧✉♠❡ ❡ ❛ ár❡❛ ❞❛ ❡s❢❡r❛ ♥♦ ❝♦♥t❡①t♦ ❡❞✉❝❛❝✐♦♥❛❧ ♥❛ ♣❡r❝❡♣çã♦ ❞❡ss❡s ♣r♦❢❡ss♦r❡s✳ ❆ ♣❡sq✉✐s❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♥❛❧✐s❛r ❝♦♠♦ é r❡❛✲ ❧✐③❛❞❛ ❛ ❛❜♦r❞❛❣❡♠ ❡ q✉❛✐s ♦s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❡❧♦s ♣r♦❢❡ss♦r❡s ♥❡ss❛ ❛❜♦r❞❛❣❡♠✳ ❖ q✉❡st✐♦♥ár✐♦ ❝♦♠♣♦st♦ ♣♦r ✶✹ q✉❡stõ❡s ✱ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ três ♣❛rt❡s✿ ❞❛❞♦s ♣❡s✲ s♦❛✐s✱ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛ ❡ ❡①♣❡r✐ê♥❝✐❛ ❞♦❝❡♥t❡✳ ❊❧❡ ❢♦✐ ❛♣❧✐❝❛❞♦ ❡♥tr❡ ♥♦✈❡♠❜r♦ ❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✸ ❝♦♠ ✸✵ ♣r♦❢❡ss♦r❡s ❡♥tr❡✈✐st❛❞♦s✭P❊✮❡♠ ◆❛t❛❧✱ ❛ três ❝❛t❡❣♦r✐❛s✿ ❘❡❞❡ Pú❜❧✐❝❛ ❋❡❞❡r❛❧✱ ❘❡❞❡ Pú❜❧✐❝❛ ❊st❛❞✉❛❧ ❡ ❘❡❞❡ Pr✐✈❛❞❛✱ s❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ♣r♦❢❡ss♦r❡s ❡♥tr❡✈✐st❛❞♦s ❞✐✈✐❞✐❞♦s ❡♠ ♣❛rt❡s ✐❣✉❛✐s ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡ss❡s s❡t♦r❡s✳ ❆ s❡❣✉✐r✱ t❡♠♦s ♦ q✉❡st✐♦♥ár✐♦✳

❆s q✉❡stõ❡s ❞❡ 01 ❛ 07 r❡❢❡r❡♠✲s❡ à ❝❛r❛❝t❡r✐③❛çã♦ ❞♦s ♣r♦❢❡ss♦r❡s✱ ❛s q✉❡stõ❡s

❞❡ 08❛ 13❢❛③❡♠ ❛ ❛♥á❧✐s❡ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t❛ ❡ ❛ q✉❡stã♦ 14❡❧✉❝✐❞❛ ❛s ❞✐✜❝✉❧❞❛❞❡s

❞♦s ♣r♦❢❡ss♦r❡s ♥❛ ❛❜♦r❞❛❣❡♠ ❞❡st❡ t❡♠❛ ♥♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛✳

I ✲ ❉❆❉❖❙ P❊❙❙❖❆■❙

01✲ ❙❡①♦✿

❉❛❞♦s ❞❛ P❡sq✉✐s❛ ❈❛♣ít✉❧♦ ✷

✭ ✮ ▼❛s❝✉❧✐♥♦ ✭ ✮ ❋❡♠✐♥✐♥♦

02✲ ❋❛✐①❛ ❡tár✐❛

✭ ✮ 20❛ 30❛♥♦s

✭ ✮ 30❛ 40❛♥♦s

✭ ✮ 40❛ 50❛♥♦s

✭ ✮ 50♦✉ ♠❛✐s

II ✲ ❋❖❘▼❆➬➹❖ ❆❈❆❉✃▼■❈❆

03 ✲ ●r❛✉ ❞❡ ❡s❝♦❧❛r✐❞❛❞❡✿

✭ ✮ ●r❛❞✉❛çã♦ ✭ ✮ ❊s♣❡❝✐❛❧✐③❛çã♦ ✭ ✮ ▼❡str❛❞♦ ✭ ✮ ❉♦✉t♦r❛❞♦

➪r❡❛ ❞❡ ❡s♣❡❝✐❛❧✐③❛çã♦✿

04✲ ❚ér♠✐♥♦ ❞❛ ❣r❛❞✉❛çã♦✿

✭ ✮ 0❛ 10❛♥♦s

✭ ✮ 10❛ 20❛♥♦s

✭ ✮ 20❛ 30❛♥♦s

✭ ✮ 30♦✉ ♠❛✐s

III ✲ ❊❳P❊❘■✃◆❈■❆ ❉❖❈❊◆❚❊

05 ✲ ❚❡♠♣♦ ❞❡ ❛t✉❛çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✿

✭ ✮ 0❛ 5 ❛♥♦s

✭ ✮ 5❛ 10❛♥♦s

✭ ✮ 10❛ 15❛♥♦s

✭ ✮ ▼❛✐s ❞❡ 15❛♥♦s

06✲ ❆ s✉❛ ❛t✉❛çã♦ é ♥❛✿

✭ ✮ ❘❡❞❡ Pú❜❧✐❝❛ ❋❡❞❡r❛❧ ✭ ✮ ❘❡❞❡ Pú❜❧✐❝❛ ❊st❛❞✉❛❧ ✭ ✮ ❘❡❞❡ P❛rt✐❝✉❧❛r

❉❛❞♦s ❞❛ P❡sq✉✐s❛ ❈❛♣ít✉❧♦ ✷

07✲ ❱♦❝ê ❥á ❞❡✉ ❛✉❧❛ s♦❜r❡ ✈♦❧✉♠❡ ❡ ár❡❛ ❞❛ ❡s❢❡r❛❄

✭ ✮ ❙✐♠ ✭ ✮ ◆ã♦

❈❛s♦ t❡♥❤❛ r❡s♣♦♥❞✐❞♦ ✏s✐♠✑ ♥❛ q✉❡stã♦ ❛♥t❡r✐♦r✱ r❡s♣♦♥❞❛ ❛s ♣❡r❣✉♥t❛s q✉❡ s❡❣✉❡♠✳ ❈❛s♦ t❡♥❤❛ r❡s♣♦♥❞✐❞♦ ✏♥ã♦✑ ✱ ♣✉❧❡ ♣❛r❛ ❛ q✉❡stã♦14✳

❊♠ r❡❧❛çã♦ ❛♦ ❡♥s✐♥♦ ❞❛ ár❡❛ ❡ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✿

08✲ ❱♦❝ê ❥á ❢❡③ ✉s♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐❄

✭ ✮ ❙✐♠ ✭ ✮ ◆ã♦

✵✾ ✲ ❱♦❝ê ❢❛③ ✉s♦ ❞❡ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦ ♥❛s s✉❛s ❛✉❧❛s❄ ✭ ✮ ❙✐♠

✭ ✮ ◆ã♦

10✲ ❈❛s♦ t❡♥❤❛ r❡s♣♦♥❞✐❞♦ s✐♠✱ q✉❡ t✐♣♦ ❞❡ ♠❛t❡r✐❛❧ ✈♦❝ê ✉t✐❧✐③❛❄

11✲ ❱♦❝ê ❢❛③ ✉s♦ s♦♠❡♥t❡ ❞❛s ❢ór♠✉❧❛sA = 4πr2 ❡

V = 4πr

3

3 ❡ r❡s♦❧✈❡ ♣r♦❜❧❡♠❛s

❞❛ ❛♣❧✐❝❛çã♦ ❞❛s ❢ór♠✉❧❛s❄ ✭ ✮ ❙✐♠

✭ ✮ ◆ã♦

❈❛s♦ t❡♥❤❛ r❡s♣♦♥❞✐❞♦ ♥ã♦✱ q✉❛❧ ♦ ♦✉tr♦ ♠ét♦❞♦❄

12✲ ❱♦❝ê ❥á ❢♦✐ q✉❡st✐♦♥❛❞♦ s♦❜r❡ ❛ ♦r✐❣❡♠ ❞❛s ❢ór♠✉❧❛s❄

✭ ✮ ❙✐♠ ✭ ✮ ◆ã♦

❈❛s♦ t❡♥❤❛ r❡s♣♦♥❞✐❞♦ s✐♠✱ q✉❡ t✐♣♦ ❞❡ q✉❡st✐♦♥❛♠❡♥t♦❄

13✲ ❱♦❝ê ❥á t❡♥t♦✉ ❞❡♠♦♥str❛r ❛s ❢ór♠✉❧❛s ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r r❡tâ♥✲

❣✉❧♦s✱ ♥♦ ❝❛s♦ ❞❡ ár❡❛s✱ ❡ ♣♦r ❝✐❧✐♥❞r♦s✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s❄

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

✭ ✮ ❙✐♠ ✭ ✮ ◆ã♦

14 ✲ ❊♠ ♣♦✉❝❛s ❧✐♥❤❛s✱ ❛♣♦♥t❡ q✉❛✐s ❛s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ❡♥s✐♥❛r ●❡♦✲

♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❡s❢❡r❛s✳

✷✳✶✳✶ ❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s

◆♦ ❣rá✜❝♦ ❛❜❛✐①♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❢❛✐①❛ ❡tár✐❛ ❞♦s ♣r♦❢❡ss♦r❡s ❡♥tr❡✈✐st❛❞♦s✳

P♦❞❡♠♦s ♣❡r❝❡❜❡r✱ ❛♥❛❧✐s❛♥❞♦ ♦ ❣rá✜❝♦✱ q✉❡ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❡stã♦ ✐♥✐❝✐❛♥❞♦ ❛ ❝❛rr❡✐r❛ ❞♦❝❡♥t❡ ❜❡♠ ❝❡❞♦✳

◆♦ ♣ró①✐♠♦ ❣rá✜❝♦✱ ♦❜s❡r✈❛♠♦s ❛ q✉❛❧✐✜❝❛çã♦ ♣r♦✜ss✐♦♥❛❧ ❞♦s ♣r♦❢❡ss♦r❡s✳

◆♦t❡♠♦s q✉❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣❡sq✉✐s❛✱ ✼✵✪ ❞♦s ♣❡sq✉✐s❛❞♦s ❥á sã♦ ♣ós✲❣r❛❞✉❛❞♦s✱ ♣♦✐s ❛ ♠❛✐♦r✐❛ ❜✉s❝❛ t♦r♥❛r✲s❡ ♣r♦✜ss✐♦♥❛❧ ♠❛✐s q✉❛❧✐✜❝❛❞♦✱ ❞❡✈✐❞♦ ❛♦ ✐♥t❡r❡ss❡ ♣❡❧❛ ♣❡sq✉✐s❛✱ ♣❡❧❛ ♠♦t✐✈❛çã♦ ❞❡ ❡st✉❞❛r ❡ ❞❡ ❡st❛r s❡♠♣r❡ ♣r♦❞✉③✐♥❞♦ ❝✐❡♥t✐✜❝❛♠❡♥t❡✳

❆♥❛❧✐s❛♥❞♦ ❛ s❡❣✉✐r ♦ t❡♠♣♦ ❞❡s❞❡ ❛ ❝♦♥❝❧✉sã♦ ❞❛ ❣r❛❞✉❛çã♦✱ ♦❜t✐✈❡♠♦s ♦s s❡❣✉✐♥t❡s ❞❛❞♦s✿

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

❊ss❡s ♣❡r❝❡♥t✉❛✐s ❞❡♠♦♥str❛♠ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ♣r♦❢❡ss♦r❡s✱ ✹✸✪ ❞❡❧❡s✱ ❡stá ❛❜❛✐①♦ ❞❡ ✶✵ ❛♥♦s ❞❡ ❝♦♥❝❧✉sã♦ ❞♦ ❝✉rs♦✱ ♦ q✉❡ ♣❡r♠✐t❡ ❡♥t❡♥❞❡r q✉❡ ❛ ♣rát✐❝❛ ❡ ❛ ❡①♣❡r✐ê♥❝✐❛ ❞❡ s❛❧❛ ❞❡ ❛✉❧❛ ❛✐♥❞❛ ❡♥❝♦♥tr❛♠✲s❡ ❡♠ ❝♦♥s♦❧✐❞❛çã♦✳

◗✉❛♥t♦ ❛ q✉❛❧✐✜❝❛çã♦ ♥❛ ár❡❛ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❛✜♥s✱ ❝♦♥s✐❞❡r❡♠♦s ♦ ❣rá✜❝♦ ❛❜❛✐①♦✿

◆♦ t♦❝❛♥t❡ ❛♦ t❡♠♣♦ ❞❡ ❛t✉❛çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ✈❡r✐✜❝❛♠♦s ♦ ❣rá✜❝♦ ❛ s❡❣✉✐r✿

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

❈♦♥s✐❞❡r❛♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ ❛❜♦r❞❛❣❡♠ ❣❡♦♠étr✐❝❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ❝♦♥✲ ❝♦r❞❛♠♦s ❝♦♠ ❬✷✵❪✱ ❛♦ ❛✜r♠❛r q✉❡ ❛ ❢❛♠✐❧✐❛r✐③❛çã♦ ❝♦♠ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❤❛❜✐❧✐❞❛❞❡s ❧✐❣❛❞❛s à ♣❡r❝❡♣çã♦ ❡s♣❛❝✐❛❧ sã♦ ❡ss❡♥❝✐❛✐s ❡♠ ✈á✲ r✐❛s s✐t✉❛çõ❡s ❡s❝♦❧❛r❡s✱ ♥♦ ❞✐❛ ❛ ❞✐❛ ❞❛s ♣❡ss♦❛s ❡ ♥♦ ❡①❡r❝í❝✐♦ ❞❛s ♠❛✐s ✈❛r✐❛❞❛s ♣r♦✜ssõ❡s✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞♦ ♣❡❧♦ ❡①❡r❝í❝✐♦ ❞❡ ❤❛✲ ❜✐❧✐❞❛❞❡s q✉❡ ❝♦♥✜❣✉r❛♠ ✉♠❛ ❡str✉t✉r❛ ❝♦♠♣♦rt❛♠❡♥t❛❧ ❛ ♣❛rt✐r ❞❛ q✉❛❧ s❡ ♣♦❞❡ ❛♣r❡❡♥❞❡r ♦ s✐❣♥✐✜❝❛❞♦ ❡ ❛s ❢✉♥çõ❡s ❞♦ ❡♥s✐♥♦ ❞❛ ❣❡♦♠❡tr✐❛✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣❡sq✉✐s❛✱ ❛ ✉t✐❧✐③❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ❛ ❞❡♠♦♥s✲ tr❛çã♦ ❞❡ ❢ór♠✉❧❛s ♣❛r❛ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s é ❝♦♥st❛t❛❞❛ ♣❡❧♦ ❣rá✜❝♦ ❛❜❛✐①♦✳

◆❛ q✉❡stã♦ s♦❜r❡ ♦ ✉s♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ✻✸✪ ❞♦s ♣r♦❢❡ss♦r❡s ❛✜r♠❛♠ ❥á t❡r❡♠ ❛❜♦r❞❛❞♦ ❡st❡ ♣r✐♥❝í♣✐♦ ❡ ✸✼✪ ❞✐ss❡r❛♠ ♥ã♦ t❡r ❢❡✐t♦ ✉s♦ ❞❡❧❡✳ ➱ ♦♣♦rt✉♥♦ ❛ss✐♥❛❧❛r q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ é ✉♠ r❡s✉❧t❛❞♦ ✐♠♣♦rt❛♥t❡ ❡ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞♦ ♣❛r❛ ❡①♣❧✐❝✐t❛r ✐♥t✉✐t✐✈❛♠❡♥t❡✱ ❞❡ ❢♦r♠❛ ❝❧❛r❛ ❡ s✐♠♣❧❡s✱ ❛s ❢ór♠✉❧❛s ❞❛s ár❡❛s ❡ ✈♦❧✉♠❡s✳

❊♠ ❬✷✸❪✱ ♦ ❛✉t♦r ❛ss✐♥❛❧❛ q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ é ❡ss❡♥❝✐❛❧✱ t❛♥t♦ ♣❛r❛ ❝❛❧❝✉❧❛r ár❡❛s q✉❛♥t♦ ♣❛r❛ ❝❛❧❝✉❧❛r ✈♦❧✉♠❡s✳ ❊♠ r❡❧❛çã♦ ❛♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✱ ♦

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

Pr✐♥❝í♣✐♦ t❡♠ s✐❞♦ ❜❡♠ ♠❛✐s ✉t✐❧✐③❛❞♦ ♣♦r ♣❡r♠✐t✐r ❛ ♦❜t❡♥çã♦ ❞❡ ❢ór♠✉❧❛s ♣❛r❛ ❝❛❧❝✉❧❛r ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ✐♥t✉✐t✐✈❛✳

❖ ❣rá✜❝♦ ❛ s❡❣✉✐r ❞✐③ r❡s♣❡✐t♦ ❛♦ ✉s♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s✳

❈♦♥st❛t❛✲s❡ ♥❛ ❛♣r❡s❡♥t❛çã♦ ❞♦s ♣❡r❝❡♥t✉❛✐s q✉❡ ♠❡t❛❞❡ ❞♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛✲ t❡♠át✐❝❛ ♥ã♦ ❞✐♥❛♠✐③❛♠ ♦ ❡♥s✐♥♦ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❛tr❛✈és ❞❛ ✐♥s❡rçã♦ ❞❡ ♠❛tér✐❛s ❝♦♥✲ ❝r❡t♦s ♥♦ ❡s♣❛ç♦ ❛❝❛❞ê♠✐❝♦✱ ❛ss✐♠ ♥ã♦ ❝♦rr❡❧❛❝✐♦♥❛♠ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❝♦♠ ❛ ✈✐✈ê♥❝✐❛ s♦❝✐❛❧ ❞♦s ❛❧✉♥♦s✳

❙♦❜r❡ ❡ss❛ ♥❡❝❡ss✐❞❛❞❡✱ ❬✷✷❪ ❢r✐s❛ q✉❡ ♦ ❡♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❛tr❛✈és ❞❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ♣♦ss✐❜✐❧✐t❛ ❛♦s ❛❧✉♥♦s ✐❞❡♥t✐✜❝❛✲ r❡♠ ❛ r❡❧❛çã♦ ❡♥tr❡ ♦ ❝♦♥t❡ú❞♦ tr❛❜❛❧❤❛❞♦ ❡ ♦s ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳

◗✉❛♥t♦ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ s❡ ✉s❛r ❛s ❢ór♠✉❧❛s ♦✉ ♥ã♦✱ ❛♥❛❧✐s❛♠♦s ❛ ✜❣✉r❛ ❛ s❡❣✉✐r✳

◆♦ q✉❡st✐♦♥❛♠❡♥t♦ s♦❜r❡ ♦ ✉s♦ s♦♠❡♥t❡ ❞❛s ❢ór♠✉❧❛s A = πr2

❡ V = 4πr

3

3 ❡

r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❛tr❛✈és ❞❛s ❛♣❧✐❝❛çõ❡s✱ ✼✸✪ ❞♦s ❡♥tr❡✈✐st❛❞♦s ❛✜r♠❛r❛♠ q✉❡ ❛♣❡♥❛s ❛s ✉t✐❧✐③❛♠✳ ❏á ✷✼✪ ❞♦s ❡♥tr❡✈✐st❛❞♦s ❞✐ss❡r❛♠ q✉❡ ✉t✐❧✐③❛♠✱ ❛❧é♠ ❞❛s r❡❢❡r✐❞❛s ❢ór♠✉❧❛s✱ ♦✉tr♦s ♠ét♦❞♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s✱ ❝♦♠♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ ✉s♦ ❞❡ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦✳

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

❖ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r r❡✢❡t❡ ❛ r❡❛❧✐❞❛❞❡ q✉❡ ♦ ❡♥s✐♥♦ ❞❡ ♠❛t❡♠át✐❝❛ t❡♠ ❡♥✲ ❢r❡♥t❛❞♦ ❤♦❥❡✱ ❡♠ q✉❡ ❛s ❞❡♠♦♥str❛çõ❡s ❡ ❥✉st✐✜❝❛t✐✈❛s ❞❡ ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ✈ê♠ s❡♥❞♦✱ ♠✉✐t❛s ✈❡③❡s✱ ❞❡✐①❛❞❛s ❞❡ ❧❛❞♦✳ ❙♦❜r❡ ❛ r❡❧❡✈â♥❝✐❛ ❞❛ ❞❡♠♦♥str❛çã♦✱ ❬✶❪ ♣♦♥t✉❛ q✉❡ ❛ ♥ã♦ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♣r✐✈❛ ♦ ❛❧✉♥♦ ❞❡ ✉♠ ❝♦♥✈❡♥✲ ❝✐♠❡♥t♦ ♣ró♣r✐♦ ❛ r❡s♣❡✐t♦ ❞❛ ✈❡r❛❝✐❞❛❞❡ ❞❡ ✉♠❛ ❛✜r♠❛çã♦✳

➱ ✐♠♣♦rt❛♥t❡ r❡✢❡t✐r q✉❡ ♦ ✉s♦ ❞❡ss❛s ❢ór♠✉❧❛s✱ s❡❣✉♥❞♦ ♦s ♣r♦❢❡ss♦r❡s ❡♥tr❡✲ ✈✐st❛❞♦s✱ ♣r♦♣õ❡ ❢❛❝✐❧✐t❛r ♦ ❡♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❛tr❛✈és ❞❡ ♠❡♠♦r✐③❛çõ❡s s✐♠♣❧❡s✱ ❜❡♠ ❝♦♠♦ ♣r♦♣♦r❝✐♦♥❛r ✉♠❛ ❛♣r❡♥❞✐③❛❣❡♠ ♠❛✐s ❢á❝✐❧ ♣❡❧♦s ❛❧✉♥♦s✳

❖✉tr♦ q✉❡st✐♦♥❛♠❡♥t♦ ❢♦✐ ❢❡✐t♦ ❡♠ r❡❧❛çã♦ à ♦r✐❣❡♠ ❞❛s ❢ór♠✉❧❛s✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ✜❣✉r❛ ❛❝✐♠❛✱ ✹✵✪ ❞♦s ❡♥tr❡✈✐st❛❞♦s r❡s♣♦♥❞❡r❛♠ q✉❡ ❥á t✐♥❤❛♠ s✐❞♦ q✉❡st✐♦♥❛❞♦s s♦❜r❡ s✉❛ ♦r✐❣❡♠ ❡ ✻✵✪ r❡s♣♦♥❞❡r❛♠ q✉❡ ♥ã♦✳

◆♦ q✉❡ s❡ r❡❢❡r❡ às ❞❡♠♦♥str❛çõ❡s ❞❛s ❢ór♠✉❧❛s ♣♦r ❛♣r♦①✐♠❛çõ❡s✱ t❡♠♦s✿

❊ss❛s r❡s♣♦st❛s ♠♦str❛♠ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❡♥tr❡✈✐st❛❞♦s ♥ã♦ ❝♦st✉♠❛ r❡❛❧✐③❛r ❞❡♠♦♥str❛çõ❡s ❞❛s ❢ór♠✉❧❛s ♣♦r ❛♣r♦①✐♠❛çõ❡s✱ ♦ q✉❡ ♣❡r♠✐t❡ ❝♦♠♣r❡❡♥❞❡r q✉❡ ♦✉✲ tr❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛s ❛♣r❡s❡♥t❛çõ❡s ❡ ❞❡♠♦♥str❛çõ❡s ❞❛s ❢ór♠✉❧❛s ♥❡♠ s❡♠♣r❡ sã♦ ❜✉s❝❛❞❛s✳

❆♣r❡s❡♥t❛çã♦ ❞❡ ❉❛❞♦s ❡ ❆♥á❧✐s❡ ❞❡ ❘❡s✉❧t❛❞♦s ❈❛♣ít✉❧♦ ✷

◆❛ q✉❡stã♦ s✉❜❥❡t✐✈❛✱ ❢♦✐ q✉❡st✐♦♥❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s s♦❜r❡ ❛s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧✲ ❞❛❞❡s ♥♦ ❡♥s✐♥♦ ❞❡ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ❡♠ ❡s♣❡❝í✜❝♦ s♦❜r❡ ❡s❢❡r❛s✳ ❖s ♣r♦❢❡ss♦r❡s ❛♣♦♥t❛r❛♠ ❛❧❣✉♠❛s ❞✐✜❝✉❧❞❛❞❡s r❡❧❡✈❛♥t❡s✱ ❝♦♥❢♦r♠❡ ♣♦❞❡ s❡r ✈✐st♦ ♥❛ tr❛♥s❝r✐çã♦ ❛❜❛✐①♦✳

✏❯♠❛ ❞❛s ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s ❡♠ s❡ ❡♥s✐♥❛r ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ é ❢❛③❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✜❣✉r❛s ❡s♣❛❝✐❛✐s ♥♦ ♣❧❛♥♦✑ ✭P❊03✮✳

❊♠ ♦✉tr❛ ❛✜r♠❛çã♦✱ ♦ P❊06 ❛ss✐♠ s❡ ♣♦s✐❝✐♦♥❛✿

✏❆ ✐♥s❡❣✉r❛♥ç❛ ❞♦s ❛❧✉♥♦s ❝♦♠ r❡❧❛çã♦ ❛♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡✉❝❧✐❞✐❛♥❛✱ ❛ ❣r❛♥❞❡ ❞✐✜❝✉❧❞❛❞❡s ❞❡ ❛❜str❛çã♦ ❡ ✈✐s✉❛❧✐③❛çã♦ tr✐❞✐✲ ♠❡♥s✐♦♥❛❧ sã♦ ❛s♣❡❝t♦s q✉❡ ❞✐✜❝✉❧t❛♠ ✉♠ ♠❡❧❤♦r ❡♥s✐♥♦✲❛♣r❡♥❞✐③❛❣❡♠ ❞❡ ❡s❢❡r❛s✳✑

❖✉tr♦ ♣r♦❢❡ss♦r✱ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s ❝♦♥t✉♥❞❡♥t❡✱ ❛✜r♠❛✿

✏❆s ♣r✐♥❝✐♣❛✐s ❞✐✜❝✉❧❞❛❞❡s q✉❡ ❡♥❝♦♥tr♦ ♥❛s ♠✐♥❤❛s ❛✉❧❛s sã♦✿ ❞❡♠♦♥s✲ tr❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ♣♦r ♠❡✐♦ ❞❡ ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s ✈✐st❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❛❧✉♥♦s ♥❛s ♠✉❧t✐♣❧✐❝❛çõ❡s ❡ ❞✐✈✐sõ❡s ❞❡ ♥ú♠❡r♦s ❞❡❝✐♠❛✐s ✭❛♣r♦①✐♠❛çã♦ ❞❡ π✮✱ ❡ ♥♦

❝á❧❝✉❧♦ ❞❡ r❛í③❡s ❝ú❜✐❝❛s q✉❛♥❞♦ ❛s q✉❡stõ❡s ❢♦r♥❡❝❡♠ ♦ ✈♦❧✉♠❡✳✑ ✭P❊

18✮

❚♦❞❛s ❛s ❞✐✜❝✉❧❞❛❞❡s ❛♣♦♥t❛❞❛s ♣❡❧♦s ♣r♦❢❡ss♦r❡s ❡♥tr❡✈✐st❛❞♦s ❞❡♥♦t❛♠ ❛ ♥❡❝❡s✲ s✐❞❛❞❡ ❞❡ ❡♥s✐♥❛r ❛ ❣❡♦♠❡tr✐❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s ❡ ❛❝❡ssí✈❡❧ ❛♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❛❧✉♥♦✳

❆ss✐♠✱ ❞✐❛♥t❡ ❞♦s r❡s✉❧t❛❞♦s ❛♣♦♥t❛❞♦s ♥❛ ♣❡sq✉✐s❛✱ ♦♣♦rt✉♥❛♠❡♥t❡ ♣♦❞❡✲s❡ ✉t✐✲ ❧✐③❛r ❛s r❡s♦❧✉çõ❡s ♠❛t❡♠át✐❝❛s ♣❛r❛ ❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ❡str✉t✉r❛❞❛s ♥ ♣r♦♣♦st❛ ❞❡ss❡ ❡st✉❞♦ ❝♦♠♦ ✉♠ ❝❛♠✐♥❤♦ ♣♦ssí✈❡❧ ❡ ✈✐á✈❡❧ ♣❛r❛ ♦ ♣r♦❢❡ss♦r ❥✉st✐✜❝❛r ❡ s✐♠♣❧✐✲ ✜❝❛r ♦ ❡♥s✐♥♦ ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

❆ ❛❧t❡r♥❛t✐✈❛ r❡s♦❧✉t✐✈❛ é ❝♦♠♣♦st❛ ❞❡ ❝♦♥❝❡✐t♦s ❡ ❞❡♠♦♥str❛çõ❡s q✉❡ sã♦ ♥♦rt❡s ❢❛❝✐❧✐t❛❞♦r❡s ♣❛r❛ ✉♠❛ ❛♣r❡♥❞✐③❛❣❡♠ s✐❣♥✐✜❝❛t✐✈❛ ❞♦s ❛❧✉♥♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❝♦♠ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❢ór♠✉❧❛s ♣❛r❛ ♦s ❝á❧❝✉❧♦s ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r r❡tâ♥❣✉❧♦s ❡ ❝✐❧✐♥❞r♦s ♣❛r❛ ❛s ár❡❛s ❡ ✈♦❧✉♠❡s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

◆♦ ❝❛♣ít✉❧♦ ❛ s❡❣✉✐r✱ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❞❡ ✈♦❧✉♠❡ ❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ✈♦❧✉♠❡✳

❈❛♣ít✉❧♦ ✸

❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❡

❈♦♥❝❡✐t♦s ●❡♦♠étr✐❝♦s

◆❡ss❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡❧❛t♦s ❞❛ ♦r✐❣❡♠ ❞❛ ●❡♦♠❡tr✐❛✱ ❞❡s❞❡ ❛ ♣ré✲❤✐stór✐❛ ❛té ❛ ❝✐✈✐❧✐③❛çã♦ ❣r❡❣❛✳ ❆❜♦r❞❛r❡♠♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ❛ ❣❡♦♠❡tr✐❛ ❞❛ ❡s❢❡r❛✳

✸✳✶ ❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛

➱ ❞✐❢í❝✐❧ ✐❞❡♥t✐✜❝❛r♠♦s ♦ ♠❛r❝♦ ✐♥✐❝✐❛❧ ❞♦ s✉r❣✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛✱ ✉♠❛ ✈❡③ q✉❡ ❤á t❡♦r✐❛s ❞✐✈❡rs❛s s♦❜r❡ ✐ss♦✳ ❙❡❣✉♥❞♦ ❬✼❪✱ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❤♦♠❡♠ ❡ ❛s ♦❜✲ s❡r✈❛çõ❡s ❞♦ ❝♦t✐❞✐❛♥♦ ♦s ✜③❡r❛♠ ❝❛♣❛③❡s ❞❡ r❡❝♦♥❤❡❝❡r ❢♦r♠❛s ❡ ✜❣✉r❛s✱ ❞❡♥tr❡ ❡❧❛s✱ ❞❡st❛❝❛♠♦s ♦s só❧✐❞♦s✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❬✸❪✱ ❛ ♦r✐❣❡♠ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡✉✲s❡ ♥❛ é♣♦❝❛ ♣ré✲❤✐stór✐❝❛✱ ❝♦♠ ❛ ❛♥á❧✐s❡ ❞❡ ♦❜❥❡t♦s ❞❛ é♣♦❝❛✳ ❆ ♣❛rt✐r ❞❡ ❡♥tã♦✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❛s ❤❛❜✐❧✐❞❛❞❡s ❞❡ r❡❝♦♥❤❡❝❡r ❢♦r♠❛s✱ ❞❡s❡♥❤♦s ❡ ✜❣✉r❛s s✐♠étr✐❝❛s✳

◆❛ ❝✐✈✐❧✐③❛çã♦ ❡❣í♣❝✐❛✱ ❞❡st❛❝❛♠✲s❡ ❞♦✐s ♣❡r❝✉rs♦r❡s✿ ❆r✐stót❡❧❡s✱ q✉❡ ❛✜r♠❛✈❛ q✉❡ ❛ ♦r✐❣❡♠ ❞❡❧❛ s❡ ❞❡✉ ♣♦r ✉♠❛ ❝❧❛ss❡ s♦❝✐❛❧ s♦❜❡r❛♥❛ q✉❡ ❞❡❞✐❝❛✈❛ t♦❞♦ ♦ s❡✉ t❡♠♣♦ ❛♦s ❡st✉❞♦s ❡ ❛♦ ❧❛③❡r❀ ❡ ❍❡ró❞♦t♦✱ q✉❡ ❛✜r♠❛✈❛ q✉❡ ❛ ♦r✐❣❡♠ ❞❛ ❝✐ê♥❝✐❛ s❡ ❞❡✉ ♣♦r ✐♥t❡r♠é❞✐♦ ❞❛s ♠❡❞✐❞❛s ❞❡ t❡rr❛s ❞❡♠❛r❝❛❞❛s ❞❡✈✐❞♦ às ❡♥❝❤❡♥t❡s ❞♦ r✐♦ ◆✐❧♦✳

❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❈❛♣ít✉❧♦ ✸

❋✐❣✉r❛ ✸✳✶✿ ❊❣✐t♦ ❆♥t✐❣♦ ❉✐s♣♦♥í✈❡❧ ❡♠✿

❁❤tt♣✿✴✴♣r♦❢❤✉❣♦❧❡♦♥❛r❞♦✳❜❧♦❣s♣♦t✳❝♦♠✳❜r✴✷✵✶✶✴✵✷✴❡❣✐t♦✲❛♥t✐❣♦✳❤t♠❧❃ ❆❝❡ss♦ ❡♠✿ ✷✵ ❞❡ ❏❛♥❡✐r♦ ❞❡ ✷✵✶✹✳

❆ ❝✐✈✐❧✐③❛çã♦ ❡❣í♣❝✐❛ s❡♠♣r❡ t❡✈❡ ❛ s✉❛ ✈✐❞❛ ❧✐❣❛❞❛ às ❞♦ r✐♦ ◆✐❧♦ ❡ ❛♦s s❡✉s ♣❡✲ rí♦❞♦s ❞❡ ❝❤❡✐❛✳ P♦r s✉❛s t❡rr❛s s❡r❡♠ ❢ért❡✐s t♦r♥♦✉✲s❡ ♣♦ssí✈❡❧ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛❣r✐❝✉❧t✉r❛ ❛ ❞❛ ❡❝♦♥♦♠✐❛✳ ❙❡❣✉♥❞♦ ❬✸❪✱ ♥❛ ❣❡♦♠❡tr✐❛ ❡❣í♣❝✐❛ ❣❛♥❤❛♠ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡ ♦s ♣❛♣✐r♦s✱ ❞♦❝✉♠❡♥t♦s q✉❡ ❝♦♥t✐♥❤❛♠ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s ❞❛ ♠❛t❡♠á✲ t✐❝❛ ❡❣í♣❝✐❛ ❞❛ é♣♦❝❛✳ ❖ P❛♣✐r♦ ❞❡ ❘❤✐♥❞ ❡ ♦ P❛♣✐r♦ ❞❡ ▼♦s❝♦✉✱ ♦s ♣r✐♥❝✐♣❛✐s ❞❛ ❣❡♦♠❡tr✐❛ ❡❣í♣❝✐❛ ♣♦ss✉❡♠ ✈ár✐♦s ♣r♦❜❧❡♠❛s ♣rát✐❝♦s✱ ❝♦♠ ê♥❢❛s❡ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦✱ ♥♦ ❝❛s♦✱ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳ ◆ã♦ ♣♦❞❡♠♦s ❞❡✐①❛r ❞❡ ♠❡♥❝✐♦♥❛r q✉❡ ♦s ❡❣í♣❝✐♦s t❛♠❜é♠ s❛❜✐❛♠ ❢❛③❡r ❝á❧❝✉❧♦s s♦❜r❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s✳

❉❡ ❛❝♦r❞♦ ❝♦♠ ❡st✉❞♦s✱ ♣❛r❛ ❛ ❝✐✈✐❧✐③❛çã♦ ❜❛❜✐❧ô♥✐❝❛✱ ❛ ❣❡♦♠❡tr✐❛ ♦r✐❣✐♥♦✉✲s❡ ❞♦ ❝á❧❝✉❧♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦s✱ ár❡❛s ❡ ✈♦❧✉♠❡s✳ ◆❡ss❛ ❝✐✈✐❧✐③❛çã♦✱ ❞❡st❛❝❛♠✲s❡ ❛s t❛❜❧❡t❛s✱ q✉❡ sã♦ tá❜✉❛s ❡♠ ❡s❝r✐t❛s ❝✉♥❡✐❢♦r♠❡s✱ ❢❡✐t❛s ❞❡ ❛r❣✐❧❛s ❡ ❜❛rr♦✱ ❡ q✉❡

❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❈❛♣ít✉❧♦ ✸

❝♦♥t✐♥❤❛♠ ❞✐✈❡rs❛s ✐♥❢♦r♠❛çõ❡s ♠❛t❡♠át✐❝❛s ❞❛ é♣♦❝❛✳ ❆ t❛❜❧❡t❛ ❞❡ P❧✐♠♣t♦♥

322✱ ❛♣r❡s❡♥t❛❞❛ ❛❜❛✐①♦✱ é ❛ ♠❛✐s ❢❛♠♦s❛ ❡ ❝♦♥té♠ ♦s t❡r♥♦s ♣✐t❛❣ór✐❝♦s q✉❡ sã♦

❢♦r♠❛❞♦s ♣♦r três ♥ú♠❡r♦s ♥❛t✉r❛✐sa✱b❡ct❛✐s q✉❡a2

+b2

=c2

✳ ❖ ♥♦♠❡ ♦r✐❣✐♥♦✉✲s❡ ❞♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s q✉❡ ❛✜r♠❛ q✉❡ s❡ ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡♥tã♦ sã♦ ✉♠ t❡r♥♦ ♣✐t❛❣ór✐❝♦✳

❋✐❣✉r❛ ✸✳✷✿ P❧✐♠♣t♦♥ ✸✷✷ ❉✐s♣♦♥í✈❡❧ ❡♠✿

❁❤tt♣✿✴✴✐s❛✇✳♥②✉✳❡❞✉✴❡①❤✐❜✐t✐♦♥s✴❜❡❢♦r❡✲♣②t❤❛❣♦r❛s✴✐t❡♠s✴♣❧✐♠♣t♦♥✲✸✷✷✴❃ ❆❝❡ss♦ ❡♠ ✿ ✷✵ ❞❡ ❏❛♥❡✐r♦ ❞❡ ✷✵✶✹✳

❙❡❣✉♥❞♦ ❬✸❪✱ ❛ ❝✐✈✐❧✐③❛çã♦ ❜❛❜✐❧ô♥✐❝❛ s❛❜✐❛ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ❞✐✈❡rs♦s só❧✐❞♦s✱ ❝♦♠♦ ❝♦♥❡s✱ ❝✐❧✐♥❞r♦s ❡ ❡s❢❡r❛s ❡♠ s✐t✉❛çõ❡s ❞❡ ❞❡♠❛r❝❛çõ❡s ❞❡ t❡rr❛s✳

❆s ❝✐✈✐❧✐③❛çõ❡s ❡❣í♣❝✐❛ ❡ ❜❛❜✐❧ô♥✐❝❛✱ ❛❧é♠ ❞❡ ♦✉tr❛s✱ ❜❡♥❡✜❝✐❛r❛♠ ❛ ❝✐✈✐❧✐③❛çã♦ ❣r❡❣❛✱ q✉❡ ❛♥t❡s s❡ ❜❛s❡❛✈❛ ♥♦ ❝á❧❝✉❧♦ ❞❡ ♠❡❞✐❞❛s✱ ♠❛s ❝♦♠ ♦ t❡♠♣♦ ♣❛ss♦✉ ❛ ❞❡❞✐❝❛r✲s❡ ♠❛✐s às ❞❡♠♦♥str❛çõ❡s ❡ ❝♦♥❝❡✐t♦s ♣r❡❝✐s♦s✳

❊♥tr❡ ❛s ✈ár✐❛s ♦❜r❛s ❡s❝r✐t❛s ♣❡❧♦s ❣r❡❣♦s✱ ♠❡r❡❝❡✉ ❞❡st❛q✉❡ ❊❧❡♠❡♥t♦s✱ ❞❡ ❊✉❝❧✐❞❡s✳ ❆ ♦❜r❛✱ q✉❡ ❢♦✐ ❜❛st❛♥t❡ ✉s❛❞❛ ♣❡❧♦s ♠❛t❡♠át✐❝♦s ✈✐♥❞♦✉r♦s✱ é ❝♦♠♣♦st❛ ❞❡13✈♦❧✉♠❡s ❡ ❞❡s❝r❡✈❡ ❛ ♠❛t❡♠át✐❝❛ ❡ ❛ ❣❡♦♠❡tr✐❛ ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ♦r❣❛♥✐③❛❞❛✳

◆♦ ▲✐✈r♦ XII✱ ❊✉❝❧✐❞❡s ❥á ❝❛❧❝✉❧❛✈❛ ✈♦❧✉♠❡s ❞❡ ❝✐❧✐♥❞r♦s✱ ❝♦♥❡s ❡ ♣✐râ♠✐❞❡s✱ ♠❛s

♥ã♦ ❝❤❡❣♦✉ ❛ ❢ór♠✉❧❛ ♠❛t❡♠át✐❝❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✱ ❛♣❡s❛r ❞❡ ❡♠ s✉❛ ♦✉tr❛ ♦❜r❛✱ ▲✉❣❛r❡s ♥❛ ❙✉♣❡r❢í❝✐❡✱ ❡❧❡ ❞❡s❝r❡✈❡r ♦s ♣r✐♠❡✐r♦s ❝♦♥❝❡✐t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡s❢ér✐❝❛✳

❆s♣❡❝t♦s ❍✐stór✐❝♦s ❞❛ ●❡♦♠❡tr✐❛ ❈❛♣ít✉❧♦ ✸

❋✐❣✉r❛ ✸✳✸✿ ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s

❉✐s♣♦♥í✈❡❧ ❡♠✿ ❁❤tt♣✿✴✴❛❧❣✉✐♠❛r❛❡s✳✇♦r❞♣r❡ss✳❝♦♠✴✷✵✶✸✴✵✽✴✶✸✴♦s✲♣♦st✉❧❛❞♦s✲❞❡✲ ❡✉❝❧✐❞❡s✲✐❞❡✐❛s✲❣❡♥✐❛✐s✲✵✷✴

❆❝❡ss♦ ❡♠✿ ✷✵ ❞❡ ❏❛♥❡✐r♦ ❞❡ ✷✵✶✹✳

❖s ❢❛t♦s ❞❛ é♣♦❝❛ r❡❧❛t❛♠ q✉❡ ❊✉❝❧✐❞❡s ❥á ✉s❛✈❛ ♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ❞❡ ❆rq✉✐✲ ♠❡❞❡s ✭287 ❛✳❈ ❛ 212 ❛✳❈✮ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✱

q✉❡ é ❞❛❞❛ ♣♦r V = cR3✱ ❡♠ q✉❡

c é ✉♠❛ ❝♦♥st❛♥t❡ ❡ R é ♦ r❛✐♦ ❞❛ ❡s❢❡r❛✳ ❊✉✲

❝❧✐❞❡s ♥ã♦ ❝♦♥s❡❣✉✐✉ ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❡ss❛ ❝♦♥st❛♥t❡✱ ♠❛s ❆rq✉✐♠❡❞❡s ❝❤❡❣♦✉ ❛ ❝♦♥❝❧✉sã♦ ♣❛r❛ ♦ ✈❛❧♦r ❞❛ ❝♦♥st❛♥t❡ ❝♦♠♦ s❡♥❞♦ 4π

3 ✳

❆rq✉✐♠❡❞❡s✱ q✉❡ s✉r❣✐✉ ❧♦❣♦ ❛♣ós ❛ ♠♦rt❡ ❞❡ ❊✉❝❧✐❞❡s✱ ❢♦✐ ♦✉tr♦ ❣❡ô♠❡tr❛ ❣r❡❣♦ ❞❡ ❞❡st❛q✉❡ ❡ ❢♦✐ ♦ ♣r✐♠❡✐r♦ q✉❡ ♣r♦✈♦✉ q✉❡ ❛ ár❡❛ ❞❡ ✉♠❛ ❡s❢❡r❛ é q✉❛tr♦ ✈❡③❡s ❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ♠á①✐♠♦ ❞❛ ♠❡s♠❛✳ ❙❡❣✉♥❞♦ ❬✸❪✱ ♥❛ ♦❜r❛ ❙♦❜r❡ ❛ ❊s❢❡r❛ ❡ ❈✐❧✐♥❞r♦✱ ♦ ❣❡ô♠❡tr❛ ❝✐t❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ❛✜r♠❛♥❞♦ q✉❡ ✧t♦❞❛ ❡s❢❡r❛ é ✐❣✉❛❧ ❛ q✉❛tr♦ ✈❡③❡s ♦ ❝♦♥❡ q✉❡ t❡♠ ❜❛s❡ ✐❣✉❛❧ ❛♦ ❝ír❝✉❧♦ ♠á①✐♠♦ ❞❛ ❡s❢❡r❛ ❡ ❛❧t✉r❛ ✐❣✉❛❧ ❛♦ r❛✐♦ ❞❛ ❡s❢❡r❛✧✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❈❛♣ít✉❧♦ ✸

✸✳✷ ❆ ◆♦çã♦ ■♥t✉✐t✐✈❛ ❞❡ ❱♦❧✉♠❡

◆❡st❡ tó♣✐❝♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ s✐❣♥✐✜❝❛❞♦ ♠❛✐s ♣r❡❝✐s♦ ♣❛r❛ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ❞❡✜♥✐r✱ ❝❧❛r❛♠❡♥t❡✱ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❙ q✉❛❧q✉❡r✳

❆ ♣r✐♠❡✐r❛ ✐❞❡✐❛ q✉❡ t❡♠♦s s♦❜r❡ ✈♦❧✉♠❡ é ❛ s✉❛ ♥♦çã♦ ✐♥t✉✐t✐✈❛✱ ♦✉ s❡❥❛✱ ❛ q✉❛♥✲ t✐❞❛❞❡ ❞❡ ❡s♣❛ç♦ ♣♦r ❡❧❡ ♦❝✉♣❛❞♦✳ ❈♦♠♦ q✉❡r❡♠♦s ♠❡❞✐r ✉♠❛ ❣r❛♥❞❡③❛ q✉❛❧q✉❡r✱ ♥♦ ❝❛s♦✱ ♦ ✈♦❧✉♠❡✱ ❞❡✈❡♠♦s ❝♦♠♣❛rá✲❧❛ ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡ ♣❛❞rã♦ q✉❡ ♥♦s s✐r✈❛ ❞❡ r❡❢❡rê♥❝✐❛✱ ♣♦r ❡①❡♠♣❧♦ ♦ ❝✉❜♦ ✉♥✐tár✐♦✱ ❝✉❥❛ ❛r❡st❛ ♠❡❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡ s❡✉ ✈♦❧✉♠❡ ♠❡❞❡ ✶✳ ❆ss✐♠✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ ❢♦r♥❡❝❡rá ❛ ♠❡❞✐❞❛ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❙ q✉❛❧q✉❡r✳

❋✐❣✉r❛ ✸✳✹✿ ❈✉❜♦ ✉♥✐tár✐♦

❖ ✈♦❧✉♠❡ ❞♦ só❧✐❞♦ ❙ s❡rá ❞❛❞♦ ♣❡❧♦ ♥ú♠❡r♦ q✉❡ ✐♥❞✐q✉❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝✉❜♦s ✉♥✐tár✐♦s q✉❡ ❝❛❜❡♠ ❡♠ ❙✱ ♣♦✐s ❛ s♦♠❛ ❞♦s ✈♦❧✉♠❡s ❞♦s ❝✉❜♦s ✉♥✐tár✐♦s ♥♦s ❞❛r✐❛ ♦ ✈♦❧✉♠❡ ❞♦ só❧✐❞♦ ❙✳

✸✳✸ ❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡

❯♠ ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r✱ é ♦ só❧✐❞♦ ❧✐♠✐t❛❞♦ ♣♦r s❡✐s ❢❛❝❡s r❡t❛♥❣✉❧❛r❡s ❝✉❥♦s ❧❛❞♦s sã♦ ❛s ❛r❡st❛s ❞♦ ♣♦❧✐❡❞r♦✳ ❖ ♣♦❧✐❡❞r♦ ✜❝❛ ❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ ❝♦♥❤❡❝❡♠♦s ❛s s✉❛s três ❛r❡st❛s ❝♦♥❝♦rr❡♥t❡s ❛ ✉♠ ♠❡s♠♦ ♣♦♥t♦✳ ❖ ❝✉❜♦ é ✉♠ ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r ❝✉❥❛s ❛r❡st❛s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛ ❡ t♦❞❛s ❛s ❢❛❝❡s sã♦ q✉❛❞r❛❞❛s✳

❯s❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ✐♥t✉✐t✐✈♦ ❞❡ ✈♦❧✉♠❡✱ ✈❛♠♦s ❞❡♠♦♥str❛r ♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❈❛♣ít✉❧♦ ✸

❚❡♦r❡♠❛ ✶ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ n é V =n3

✳

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ♦s ❝❛s♦s ❡♠ q✉❡ ❛ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛né ✐♥t❡✐r❛✱

r❛❝✐♦♥❛❧ ❡ ✐rr❛❝✐♦♥❛❧✳

❉❡ ❢❛t♦✱ s❡ t❡♠♦s ✉♠ ❝✉❜♦ A ❝✉❥❛ ❛r❡st❛ ♠❡❞❡ n ✉♥✐❞❛❞❡s✱ s❡♥❞♦ n ✉♠ ♥ú✲

♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ♦ ❝✉❜♦ ❡♠n3

❝✉❜♦s ✉♥✐tár✐♦s ❥✉st❛♣♦st♦s✳ ❆ss✐♠✱ ♦ s❡✉ ✈♦❧✉♠❡ s❡rá V =n3

✳

◆♦t❡ ❛❣♦r❛ q✉❡✱ s❡ n ❢♦r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞❛ ❢♦r♠❛ 1

q ✱ ♦♥❞❡ q é ✉♠ ♥ú♠❡r♦

✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❞❡❝♦♠♣♦♥❞♦ ❛s ❛r❡st❛s ❞♦ ❝✉❜♦ ✉♥✐tár✐♦ ❡♠ q♣❛rt❡s ✐❣✉❛✐s✱ ❡❧❡ ✜❝❛

❞✐✈✐❞✐❞♦ ❡♠ q3

❝✉❜♦s ♠❡♥♦r❡s ❞❡ ❛r❡st❛ 1

q✳ ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s

V_{❝✉❜♦ ✉♥✐tár✐♦} = 1 =q3.V_{❝✉❜♦ ♠❡♥♦r} ⇒V_{❝✉❜♦ ♠❡♥♦r} = 1

q3 =

1

q

3

P♦rt❛♥t♦✱ ♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦ ❞❡ ❛r❡st❛ 1

q é 1 q 3 ✳

◗✉❛♥❞♦ ❞❛❞♦ ✉♠ ❝✉❜♦ ❝✉❥❛ ❛r❡st❛ é ♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ n ❞❛ ❢♦r♠❛ n = p

q ✱

♣♦❞❡♠♦s ❞❡❝♦♠♣♦r ❝❛❞❛ ✉♠❛ ❞❡ s✉❛s ❛r❡st❛s ❡♠ p ♣❛rt❡s ✐❣✉❛✐s✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s

❝♦♠ ❝♦♠♣r✐♠❡♥t♦ 1

q✳ ❉❡st❡ ♠♦❞♦✱ ♦ ❝✉❜♦ ✜❝❛rá ❞✐✈✐❞✐❞♦ ❡♠ p

3 ❝✉❜♦s ❥✉st❛♣♦st♦s✱

❝❛❞❛ ✉♠ ❝♦♠ ❛r❡st❛ 1

q ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱ ❝❛❞❛ ✈♦❧✉♠❡ ❞♦s ❝✉❜♦s ♠❡♥♦r❡s é

1

q3✳

V_❝✉❜♦ =p3.1 q3 =

p3 q3 =

p q

3

P♦rt❛♥t♦✱ s❡ ❛ ❛r❡st❛ ❞❡ ✉♠ ❝✉❜♦ t❡♠ ❝♦♠♦ ♠❡❞✐❞❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧n✱ ❡♥tã♦

♦ s❡✉ ✈♦❧✉♠❡ é ✐❣✉❛❧ ❛ n3✳

❙♦❜ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 1✳ P♦ré♠✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♣rát✐❝♦✱

♥ã♦ é ♣♦ssí✈❡❧ ♦❜t❡r ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ❝♦♠♦ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛ ❞❡ ✉♠ ❝✉❜♦ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✐♥str✉♠❡♥t♦ ❞❡ ♠❡❞✐çã♦✱ ♠❛s s❡♠♣r❡ ❝♦♥s❡❣✉✐r❡♠♦s ♦❜t❡r ✈❛❧♦r❡s ❛♣r♦①✐♠❛❞♦s ❛ ✐rr❛❝✐♦♥❛✐s✳

❖❜s❡r✈❡ ❛❣♦r❛ q✉❡ s❡ n é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✱ ✈❛♠♦s ♣r♦✈❛r q✉❡ ♦ ✈♦❧✉♠❡ ❞❡

✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ n é ❞❛❞♦ ♣♦r V =n3✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❈❛♣ít✉❧♦ ✸

❚♦♠❡ q✉❛❧q✉❡r ♥ú♠❡r♦ x < n3

✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r < n tã♦ ♣ró①✐♠♦

❞❡ n q✉❛♥t♦ ❞❡s❡❥❛♠♦s✱ ♦✉ s❡❥❛ x < r3 < n3

✳

❉❡st❛ ♠❛♥❡✐r❛✱ ♦ ❝✉❜♦ A✱ ❝✉❥❛ ❛r❡st❛ é n✱ ❝♦♥té♠ ✉♠ ❝✉❜♦ B✱ ❝✉❥❛ ❛r❡st❛ é ♦

♥ú♠❡r♦ r❛❝✐♦♥❛❧r✱ ❡♥tã♦✱ VB < VA✳ ▼❛s ❝♦♠♦ ♦ ✈♦❧✉♠❡ ❞❡B ér 3

✱ ♣♦✐sré r❛❝✐♦♥❛❧✱

❝♦♥❝❧✉í♠♦s q✉❡ x < VA ✳

❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ t♦♠❡ q✉❛❧q✉❡ry > n3

✳ ❊①✐st❡ ✉♠sr❛❝✐♦♥❛❧✱ s❡♥❞♦s > n✱ tã♦

♣ró①✐♠♦s ❞❡ n q✉❛♥t♦ ❞❡s❡❥❛♠♦s✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ n3 < s3

< y✳

❊♥tã♦✱ ✉♠ ❝✉❜♦ ❈ ❝✉❥❛ ❛r❡st❛ é s r❛❝✐♦♥❛❧ ❝♦♥té♠ ♦ ❝✉❜♦ ❞❡ ❛r❡st❛ n ❡ ❛ss✐♠ VA< VC < y✳ ▲♦❣♦✱ VA< y✳

❉❡ ❢❛t♦✱ s❡ x ❡ y sã♦ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r✱ t❛✐s q✉❡ x < n3

< y✱ t❡♠♦s x < VA < y✳ ❙✉♣♦♥❤❛ q✉❡ VA 6=n

3

✱ ❞✐❣❛♠♦s VA > n 3

✳ ❉❡ss❛ ❢♦r♠❛✱ s❡r✐❛ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ①✱ ② ❝♦♠ x < n3

< y < VA ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r✳

▲♦❣♦ ❞❡✈❡♠♦s t❡r VA = n 3

✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ s✉♣♦♥❤❛ q✉❡ VA 6= n 3

✱ ❞✐❣❛♠♦s

VA< n 3

✳ ❆ss✐♠✱ s❡r✐❛ ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ✈❛❧♦r❡s ♣❛r❛ ①✱ ② ❝♦♠ x > n3

> y > VA ♦

q✉❡ ❞✐✈❡r❣❡ ❞♦ q✉❡ ❛❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r✳ P♦rt❛♥t♦ ❞❡✈❡♠♦s t❡r VA=n 3

✳

❚❡♦r❡♠❛ ✷ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡s r❡❛✐s x✱ y ❡ z

é V =xyz✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❝✉❥❛s ❞✐♠❡♥sõ❡s s❡❥❛♠ ♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐sx✱y❡z✳ P♦❞❡♠♦s s❡♠♣r❡ ❡s❝r❡✈❡r ♦s ♥ú♠❡r♦s ❡♠ ✉♠ ♠❡s♠♦ ❞❡♥♦♠✐♥❛❞♦r

❡✱ ♣♦rt❛♥t♦✱ t❛✐s ❞✐♠❡♥sõ❡s sã♦ r❡♣r❡s❡♥t❛❞❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r x

q✱ y q ❡

z

q✱ ❡♠ q✉❡ x✱ y✱ z ❡q sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳

❆♦ ❞❡❝♦♠♣♦r ❛s ❛r❡st❛s x✱ y✱ z ❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ a✱ b ❡ c✱ s❡❣♠❡♥t♦s ❞❡

❝♦♠♣r✐♠❡♥t♦s 1

q ❝❛❞❛✱ ♦ ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r ✜❝❛rá ❞✐✈✐❞✐❞♦ ❡♠abc❝✉❜♦s ❥✉st❛♣♦st♦s

❞❡ ❛r❡st❛s 1

q ❡ ✈♦❧✉♠❡s

1

q3 ❝❛❞❛✳ ▲♦❣♦✱

V_{♣❛r❛❧❡❧❡♣í♣❡❞♦} =abc.

1 q 3 = a q. b q. c q

P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❞✐♠❡♥sõ❡s r❛❝✐✲ ♦♥❛✐s x✱ y ❡ z é ❞❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ s✉❛s ❞✐♠❡♥sõ❡s✱ ♦✉ s❡❥❛✱ V =xyz✳

❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ P ❝✉❥❛s ❞✐♠❡♥sõ❡s sã♦ ♥ú♠❡r♦s x✱ y ❡ z✱ ❡♠ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡ s✉❛s ❞✐♠❡♥sõ❡s s❡❥❛ ✐rr❛❝✐♦♥❛❧✳ ◆❡ss❡ ❝❛s♦✱ ♦ ✈♦❧✉♠❡

t❛♠❜é♠ s❡rá ❞❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ s✉❛s ❞✐♠❡♥sõ❡s✳

❖ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❈❛♣ít✉❧♦ ✸

❚♦♠❡ ✉♠cq✉❛❧q✉❡r ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡c < xyz✳ ❊♥tã♦✱ ❝♦♥s❡❣✉✐r❡♠♦s ❡♥❝♦♥tr❛r

♥ú♠❡r♦s r❛❝✐♦♥❛✐s r✱ s ❡ t✱ ❞❡ ♠♦❞♦ q✉❡r < x✱ s < y ❡ t < z tã♦ ♣ró①✐♠♦s q✉❛♥t♦

q✉❡r❡♠♦s ❞❡ x✱ y ❡ z✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ c < rst < xyz✳ ❉❡ ❢❛t♦✱ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ P ❝♦♥té♠ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ T1 ♠❡♥♦r✱ ❝✉❥❛s ❛r❡st❛s ♠❡❞❡♠ r✱ s ❡ t✳ ▲♦❣♦✱ VP > VT1 =rst=> c ✱ ♦✉ V_P > c ✳

❯s❛♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦✱ t♦♠❛♠♦s ✉♠dq✉❛❧q✉❡r✱ t❛❧ q✉❡d > xyz✳ P♦❞❡♠♦s

❡♥❝♦♥tr❛r ♥ú♠❡r♦s r❛❝✐♦♥❛✐su✱v ❡wt❛✐s q✉❡u > x✱v > y ❡w > z✱ tã♦ ♣ró①✐♠♦s ❞❡ x✱y❡z q✉❛♥t♦ ❞❡s❡❥❛r♠♦s✱ ❞❡ ♠♦❞♦ q✉❡xyz < uvw < d✳ ❆ss✐♠✱ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦P

❡stá ❝♦♥t✐❞♦ ❡♠ ✉♠ ♦✉tr♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦T2✱ ❝✉❥❛s ❞✐♠❡♥sõ❡s sã♦u✱v ❡w✳ P♦rt❛♥t♦✱ VP < VT2 = uvw < d✱ ♦✉ s❡❥❛✱ V_P < d✳ ❆ss✐♠✱ s❡ c < xyz < d✱ ❡♥tã♦ c < V_p < d✱

❧♦❣♦ Vp =xyz✳

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡✜♥✐çã♦ ❣❡r❛❧ ❞❡ ✈♦❧✉♠❡✳

❉❡✜♥✐çã♦ ✸ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ é ♦ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦ ❝✉❥❛s ❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛ sã♦ ❛s s♦♠❛s ❞♦s ✈♦❧✉♠❡s ❞♦s ♣♦❧✐❡❞r♦s r❡t❛♥❣✉❧❛r❡s ♥❡❧❡ ❝♦♥t✐❞♦s✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ q✉❛♥t♦ ♠❛✐s ♣♦❧✐❡❞r♦s r❡t❛♥❣✉❧❛r❡sP ❛❝r❡s❝❡♥t❛r♠♦s ❡ ✐♥s✲

❝r❡✈❡r♠♦s ❡♠ S✱ ♦❜t❡r❡♠♦s ✉♠ ♥♦✈♦ ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r P′_{✱ ♠❛✐♦r ❞♦ q✉❡ P✱ ❡}

❛ss✐♠ ♦ ✈♦❧✉♠❡ P′ _{s❡rá ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❜❡♠ ♠❡❧❤♦r ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞♦ só❧✐❞♦ S✳}

■ss♦ ♥♦s ❞✐③ q✉❡ VS ≥ VP ♣❛r❛ q✉❛❧q✉❡r ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r P ❝♦♥t✐❞♦ ❡♠ S✳

❆✐♥❞❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ s❡ t é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ t❛❧ q✉❡ t < VS✱ é ♣♦ssí✈❡❧

❡♥❝♦♥tr❛r♠♦s ✉♠ ♦✉tr♦ ♣♦❧✐❡❞r♦ r❡t❛♥❣✉❧❛r T✱ ❝♦♥t✐❞♦ ❡♠ S✱ q✉❡ ✈❛❧❡ ❛ r❡❧❛çã♦ t < VT < VS✳

❋✐❣✉r❛ ✸✳✺✿ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❝♦♥t✐❞♦s ❡♠ ❙

❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❈❛♣ít✉❧♦ ✸

❆ ❡①♣r❡ssã♦ ✧❛♣r♦①✐♠❛çõ❡s ♣♦r ❢❛❧t❛✧❛♣❡♥❛s ❢❛③ s❡♥t✐❞♦ r✐❣♦r♦s❛♠❡♥t❡✱ s❡ ✉s❛r✲ ♠♦s ❛ ♥♦çã♦ ❞❡ ❧✐♠✐t❡✱ ❢✉♥❞❛♠❡♥t❛❧ ♥♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧✱ ❛ q✉❛❧ ♥ã♦ ❢❛③ ♣❛rt❡ ❞♦ ❝✉rrí❝✉❧♦ ✉s✉❛❧ ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦✳ P♦r ✐ss♦✱ ❛❜♦r❞❛♠♦s ❡ss❛ ❡①♣r❡ssã♦ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ❛❝r❡❞✐t❛♥❞♦ q✉❡ ❡st❛ ❛❜♦r❞❛❣❡♠ é s✉✜❝✐❡♥t❡✳

❖s ♣♦❧✐❡❞r♦s r❡t❛♥❣✉❧❛r❡s ♥❛ ❞❡✜♥✐çã♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ♣♦❞❡♠ s❡r s✉❜s✲ t✐t✉í❞♦s ♣♦r ♦✉tr♦s só❧✐❞♦s ❝✉❥♦s ✈♦❧✉♠❡s ❝♦♥❤❡❝❡♠♦s✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✈❛♠♦s ❡♥✲ ❝♦♥tr❛r ❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ✉s❛♥❞♦ ❛♣r♦①✐♠❛çõ❡s ♣♦r ❝✐❧✐♥❞r♦s ♥❡❧❛ ❝♦♥t✐❞♦s✳

✸✳✹ ❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✲

✈❛❧✐❡r✐

❖ ❡♥s✐♥♦ ❞❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ é ♥♦r♠❛❧♠❡♥t❡ ❢❡✐t♦ ✉s❛♥❞♦✲s❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❆❜♦r❞❛r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ❡ss❡ ♣r✐♥❝í♣✐♦✱ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ♦❜s❡r✈❛♥❞♦ ♦ s❡✲ ❣✉✐♥t❡ ❡①❡♠♣❧♦✳

❈❛s♦ s❡❥❛♠ ❝♦❧♦❝❛❞❛s ❡♠ ❝✐♠❛ ❞❡ ✉♠❛ ♠❡s❛ ❛❧❣✉♠❛s ♠♦❡❞❛s ♣❡r❢❡✐t❛♠❡♥t❡ ❛rr✉♠❛❞❛s✱ ❡❧❛s ❢♦r♠❛rã♦ ✉♠ ❝✐❧✐♥❞r♦✳ ❊♠♣✉rr❛♥❞♦ ❧❛t❡r❛❧♠❡♥t❡ ❛s ♠♦❡❞❛s q✉❡ ♦ ❝♦♠♣õ❡ ♦ ❝✐❧✐♥❞r♦✱ ♦❜t❡r❡♠♦s ✉♠ ❝✐❧✐♥❞r♦ ♦❜❧íq✉♦ ❡✱ s❡ ❛❧t❡r❛r♠♦s ♠❛✐s ✉♠ ♣♦✉❝♦✱ t❡r❡♠♦s ❛✐♥❞❛ ♦✉tr♦s só❧✐❞♦s✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ❛ s❡❣✉✐r✿

❋✐❣✉r❛ ✸✳✻✿ ◆♦çã♦ ■♥t✉✐t✐✈❛ ❞❡ ❱♦❧✉♠❡s

❖s ✈♦❧✉♠❡s ❞❡ss❡s três só❧✐❞♦s s❡rã♦ ✐❣✉❛✐s✱ ♣♦✐s ♥ã♦ ❤♦✉✈❡ ❛❧t❡r❛çã♦ ❞❛ q✉❛♥✲ t✐❞❛❞❡ ❞❡ ♠♦❡❞❛s✳ ❙❡❣✉♥❞♦ ❬✸❪✱ ❡ss❡ ❢❛t♦ ♣♦❞❡ s❡r ❥✉st✐✜❝❛❞♦ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❈❛♣ít✉❧♦ ✸

❉❡✜♥✐çã♦ ✹ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

I✳ ❙❡ ❞♦✐s só❧✐❞♦s sã♦ t❛✐s q✉❡ t♦❞♦ ♣❧❛♥♦ s❡❝❛♥t❡ ❛ ❡❧❡s ❡ ♣❛r❛❧❡❧♦ ❛ ✉♠ ♣❧❛♥♦ ❞❛❞♦

❞❡t❡r♠✐♥❛ ♥♦s só❧✐❞♦s s❡❝çõ❡s ♣❧❛♥❛s ❝✉❥❛ r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s é ❝♦♥st❛♥t❡✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ ♦s ✈♦❧✉♠❡s ❞❡ss❡s só❧✐❞♦s é ❛ ♠❡s♠❛ ❝♦♥st❛♥t❡✳

❋✐❣✉r❛ ✸✳✼✿ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✈♦❧✉♠❡s

❉❡✜♥✐çã♦ ✺ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

II✳ ❙❡ ❞✉❛s ♣♦rçõ❡s ♣❧❛♥❛s sã♦ t❛✐s q✉❡ t♦❞❛ r❡t❛ s❡❝❛♥t❡ ❛ ❡❧❛s ❡ ♣❛r❛❧❡❧❛ ❛ ✉♠❛ r❡t❛

❞❛❞❛ ❞❡t❡r♠✐♥❛ ♥❛s ❞✉❛s ♣♦rçõ❡s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❝✉❥❛ r❛③ã♦ é ❝♦♥st❛♥t❡✱ ❡♥tã♦ ❛ r❛③ã♦ ❡♥tr❡ ❛s ár❡❛s ❞❡ss❛s ♣♦rçõ❡s é ❛ ♠❡s♠❛ ❝♦♥st❛♥t❡✳

❋✐❣✉r❛ ✸✳✽✿ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ár❡❛s

❯s❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❛ s❡❣✉✐r✱ ✐r❡♠♦s ❞❡❞✉③✐r ❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳

❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❈❛♣ít✉❧♦ ✸

❉❡✜♥✐çã♦ ✻ ❖ só❧✐❞♦ ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ❞♦✐s ❝♦♥❡s✱ t❡♥❞♦ ❝♦♠♦ ❜❛s❡s ❛s ❞♦ ❝✐❧✐♥❞r♦ ❡ s❡♥❞♦ ❙ ♦ ✈ért✐❝❡ ❝♦♠✉♠ ❛♦s ❝♦♥❡s✱ é ❞❡♥♦♠✐♥❛❞♦ ❝❧é♣s✐❞r❛✳

❉❡✜♥✐çã♦ ✼ ❖ só❧✐❞♦ ❢♦r♠❛❞♦ ♣❡❧❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❞♦ ❝✐❧✐♥❞r♦ ❡ ❡①t❡r♥❛ à ❝❧é♣s✐✲ ❞r❛✭✈❡r ❞❡✜♥✐çã♦ ✻✮✱ é ❞❡♥♦♠✐♥❛❞♦ ❛♥t✐❝❧é♣s✐❞r❛✳

❊♠ s❡❣✉✐❞❛✱ ✉t✐❧✐③❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ ❛s ❉❡✜♥✐çõ❡s ✻ ❡ ✼ ♣❛r❛ ❞❡❞✉③✐r ❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ ❘✳

▲❡♠❛ ✼✳✶ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ R é ❞❛❞♦ ♣♦r V = 4πR

3

3 ✳

❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ❝♦♥s✐❞❡r❛r✿ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦ ❞❡ r❛✐♦ R ❡ ❛ ❛❧t✉r❛

2R✱ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ R✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ ❡s❢❡r❛ ❡ ♦ ❝✐❧✐♥❞r♦ ❡st❡❥❛♠ ❡♠ ✉♠

♠❡s♠♦ ♣❧❛♥♦ α ❡ ❞♦✐s ❝♦♥❡s ❞❡ r❛✐♦sR ❡ ❛❧t✉r❛ R✱ ❛♠❜♦s ❝♦♠ ✈ért✐❝❡ ♥♦ ❝❡♥tr♦ ❞♦

❝✐❧✐♥❞r♦✱ ❝✉❥❛s ❜❛s❡s s❡❥❛♠ ❛s ❜❛s❡s ❞♦ ❝✐❧✐♥❞r♦ ✭❝❧é♣s✐❞r❛✮✳

❋✐❣✉r❛ ✸✳✾✿ ❱♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ♣♦r ❈❛✈❛❧✐❡r✐

❚r❛ç❛♠♦s ✉♠ ♣❧❛♥♦ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♣❛r❛❧❡❧♦ ❛α ❡ q✉❡ ✐♥t❡r❝❡♣t❛ ♦s só❧✐❞♦s ❛

✉♠❛ ♠❡s♠❛ ❛❧t✉r❛ R+h✳

❯s❛♠♦s ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ár❡❛ ❞❡ ✉♠❛ s❡❝çã♦ ❡s❢ér✐❝❛ é ❞❛❞❛ ♣♦r

A_{s❡❝çã♦ ❡s❢ér✐❝❛} =πR2−πh2

◆♦t❛♠♦s✱ ❛ss✐♠✱ q✉❡ ❛ ár❡❛ ❞❛ s❡çã♦ q✉❡ ✐♥t❡r❝❡♣t❛ ♦ ❝✐❧✐♥❞r♦ ❡ ♦ ❝♦♥❡ é ✐❣✉❛❧ ❛ ár❡❛ ❞❡ ✉♠❛ ❝♦r♦❛ ❝✐r❝✉❧❛r ❞❡ r❛✐♦s R ❡ h (R > h)✱ ♦✉ s❡❥❛✱ é ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛

ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ R ❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❞❡ r❛✐♦ h✱ ❡♥tã♦✿

A_{❝♦r♦❛ ❝✐r❝✉❧❛r} =πR2−πh2

❖ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ s❡❣✉♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❈❛♣ít✉❧♦ ✸

❉❡ ❢❛t♦✱ ♦ ♣❧❛♥♦ ♣❛r❛❧❡❧♦ ❛♦ ♣❧❛♥♦ α ❞❡t❡r♠✐♥❛rá s❡çõ❡s q✉❡ ♣♦ss✉❡♠ ❛ ♠❡s♠❛

ár❡❛✳ ❊♥tã♦✱ ❝♦♠♦ ❛s ❛❧t✉r❛s ❞♦s só❧✐❞♦s sã♦ ✐❣✉❛✐s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♦s s❡✉s ✈♦❧✉♠❡s s❡rã♦ ✐❣✉❛✐s✳ ❆ss✐♠✱

V_{❡s❢❡r❛} = V_{❛♥t✐❝❧é♣s✐❞r❛}

= V_{❝✐❧✐♥❞r♦}−2V_{❝♦♥❡s}

= A_❜❛s❡.H −2.1

3A❜❛s❡.H

= πR2.2R− 2

3πR

2 .R

= 2πR3− 2

3πR

3

= 4πr

3

3

P♦rt❛♥t♦✱ ❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❞❡ r❛✐♦R é 4πR 3

3 ✳

❈❛♣ít✉❧♦ ✹

➪r❡❛s ❡ ❱♦❧✉♠❡s ❯s❛♥❞♦

❆♣r♦①✐♠❛çõ❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❞❡❞✉③✐r ❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ❡ ♣❛r❛ ❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛✳ ❚❛✐s ❞❡❞✉çõ❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✾❪✳

✹✳✶ ❱♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❛tr❛✈és ❞❡ ❛♣r♦①✐♠❛çõ❡s ♣♦r

❝✐❧✐♥❞r♦s

❱❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ♣♦♥t♦ O ♥♦ ❡s♣❛ç♦ ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ R > 0✳ ❆ss✐♠✱

❞❡✜♥✐♠♦s ❝♦♠♦ s✉♣❡r❢í❝✐❡ ❡s❢ér✐❝❛ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ R ♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s ❞♦

❡s♣❛ç♦ ❝✉❥❛ ❞✐stâ♥❝✐❛ ❛♦ ❝❡♥tr♦ O é ✐❣✉❛❧ ❛♦ r❛✐♦ R✳

❋✐❣✉r❛ ✹✳✶✿ ❙✉♣❡r❢í❝í❡ ❊s❢ér✐❝❛